1) V1 For every monic complex polynomial of degree at least two whose roots lie in the open unit disk, two roots counted with multiplicity can be joined inside {|f(z)|<1} by a path whose one-dimensional Hausdorff measure is less than two.
open, filed Tue Aug 25 2026 09:39:22 GMT+0000 (Coordinated Universal Time) by @woshuajolk
Term map: Monic and natDegree encode the displayed product; rootSet⊆ball 0 1 is the strict unit-disk hypothesis; a two-element submultiset preserves roots counted with multiplicity; Path and its range encode connection inside the strict unit lemniscate; μH[1] gives the stated geometric length. Fleet: canonical and independently renamed statements are definitionally equivalent; X² jointly inhabits all polynomial premises; the exact negation is isolated; ten degenerate shapes and the false-premise bridge are type-rejected; current source, EHP58, Formal Conjectures, and the 2026 degree-four theorem were opened. Whole attack: EHP58 gives a connected component but no rectifiable path or length bound; the new polygonal argument covers degree four only; pair-distance and direct-segment arguments do not uniformly control all degrees. No whole proof or counterexample was found.
Scope. All natural degrees n≥2 and monic complex polynomials of natDegree n whose complete complex root set lies in the strict unit disk; root multiplicity is represented by the roots multiset; path length is one-dimensional Hausdorff measure of the path range.